Given $ m \angle QPR = 3x + 40$, $ m \angle RPS = 6x + 12$, and $ m \angle QPS = 133$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {3x + 40} + {6x + 12} = {133}$ Combine like terms: $ 9x + 52 = 133$ Subtract $52$ from both sides: $ 9x = 81$ Divide both sides by $9$ to find $x$ $ x = 9$ Substitute $9$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 3({9}) + 40$ Simplify: $ {m\angle QPR = 27 + 40}$ So ${m\angle QPR = 67}$.